Call the headquarters of Alcor in Scottsdale, Arizona, and you are greeted by a recorded message. “If you would like to report the death or near-death of an Alcor member,” says a chirpy midwestern voice, “please press two.”

The Alcor Life Extension Foundation – to give it its full title – has an unexceptional grey concrete exterior that resembles a regional bank branch. Inside, however, are the bodies or brains of 138 dead people, stored in vats of liquid nitrogen in the hope that, at some point in the future, advances made in science will be capable of bringing them back to life.

This is cryonics – the preservation of animals and humans at extremely low temperatures. And in America, business is booming. Last month, Alcor took receipt of its 138th patient: Du Hong, a Chinese science-fiction writer who died of pancreatic cancer at the age of 61 and whose family contacted Alcor shortly before her death to have her brain preserved.

Brain-freezing starts at $100,000 and is cheaper than the full-body option, which costs more than twice that amount. Alcor, which describes itself as a not-for-profit organisation, insists that all fees go directly back into running costs.

It might sound like something lifted from the pages of a bad space-age fantasy, but cryonics advocates insist the possibility of eternal life is getting ever closer. “A hundred years ago, if a person’s heart stopped, they were dead,” says Dennis Kowalski, the president of the Cryonics Institute, which also specialises in cryonic preservation.

“Today, we bring those people back to life with a defibrillator. What changed? Science, medicine.”

Cryonics involves cooling a legally dead person to liquid nitrogen temperature where all physical decay essentially stops. The goal is to preserve tissue, organs and brain as perfectly as possible.

“Essentially, the concept is to ‘buy time’ until technology catches up and is able to fully repair and restore the human body,” explains the Cryonics Institute website.

Quite what form that restoration will take is a moot point. But there have been some notable recent advances. In April, the journal Nature Methods published a protocol by Dr Shawn Mikula from the Max Planck Institute for Medical Research in Heidelberg, which described his use of a solvent to preserve an entire mouse brain at a high enough cellular quality to enable it to be electromagnetically scanned.

If a brain can be scanned, it can, theoretically, be mapped out and uploaded on to a computer, opening up the possibility of a digital resurrection, if not a physical one.

Still, it’s a giant unknown. If, hypothetically, we imagine Du Hong’s brain could be brought back to life, would she exist in the same form? Would she still be able to speak Mandarin?

“Oh jeez,” says Kowalski. There is a long pause. “Here’s what I believe: I believe every single person we have here can be brought back as far as their DNA is concerned. How much of the mind we can save is a matter of variables, like: did we get to the patient right away? The sooner you get to the person, the more likely you are to have their mind.”

Kowalski, who works as a paramedic (his role at the Cryonics Institute is unpaid), compares it to someone suffering a stroke: the quicker the intervention, the more hope there is of a full recovery. But he admits much of this is just guesswork and blind faith.

Isn’t it all a bit, well, weird?

“Before the first heart transplant, everyone thought that was a bit weird too,” he says. “When you think about it, you think Frankenstein, right? Yet that’s exactly what people felt about organ transplantations and look how many lives have been saved and enriched by that.”

In the United States, the desire to live for ever (and to look good while doing it) has resulted both in a surge of interest in cryonics and in a booming anti-ageing industry. Spending on anti-ageing products is expected to reach $292bn by 2015; $11bn was spent on cosmetic treatments in 2012, according to the American Society of Plastic Surgeons – a 5.5% rise on the previous year.

“We want to stay alive as long as possible,” says Dave Kekich, the founder of the Maximum Life Foundation, which funds research into curing age-related diseases. “But if that doesn’t work, we want a plan B. That’s cryonics.”

In fact, Kekich has signed up for Alcor membership and intends to have his body frozen when he dies. He is even speaking at an Alcor conference this weekend, an annual event taking place in Scottsdale attended by up to 200 people. Other speakers include Alcor’s president and CEO Max More and his wife, the fantastically named Natasha Vita More.

I try to speak to Max More, but no interview is forthcoming despite several emails and calls. Marji, the Alcor office administrator (it’s her voice on the answerphone), says he’s too busy preparing for the conference.

More was born in Bristol and studied PPE at St Anne’s, Oxford, before moving to America in 1987 and getting heavily involved in cryonics and “transhumanism” – a movement devoted to defeating mortality and becoming “beyond human”.

He describes himself as “an internationally acclaimed strategic futurist” and his personal website states that he started “a personal life extension regimen in his early teens”. More is now 51, so it seems to be going pretty well.

And Alcor, too, is gaining more mainstream attention. Last month, it was the subject of an extensive New York Times article that told the harrowing story of Kim Suozzi, a 23-year-old woman with terminal cancer who chose to preserve her brain and who raised the necessary funds through online donations.

The majority of patients choose to take out life insurance policies naming Alcor or the Cryonics Institute as their beneficiary. Some are wealthy enough to cover their own costs.

Kowalski says that there is no typical patient – the institute currently has 135, ranging from young children all the way up to people in their 90s. They come from all over the world: the biggest number are from the United States, closely followed by the UK, Canada, Australia and Germany. They include professors, chefs, secretaries, students and even beloved pets.

But can we really ever be defrosted back to life? The science writer Michael Shermer has compared the idea to thawing out a can of frozen strawberries that, when defrosted, will simply turn into runny mush.

The neurosurgeon Jiang Jiyao, who has carried out brain surgery at low temperatures, also has doubts. “I cannot imagine what will happen to the human brain at that ultra-cold temperature,” he told a Chinese news site in September.

“Brain cells are extremely vulnerable and without adequate supply of blood and oxygen, irreversible damage can be caused in just four to six minutes under room temperature.”

And Michael Hendricks, a neuroscientist and assistant professor of biology at McGill University in Montreal, insists the technology required “does not yet exist even in principle”.

“Simulation, if it were to occur, would result in a new person who is like you but whose conscious experience you don’t have access to,” he wrote last month in the MIT Technology Review.

“That means that any suggestion that you can come back to life is simply snake oil … [it is] an abjectly false hope that is beyond the promise of technology and is certainly impossible with the frozen, dead tissue offered by the ‘cryonics’ industry. Those who profit from this hope deserve our anger and contempt.”

Kowalski is unruffled. Either way, he insists it’s worth taking the risk: “It’s a self-funded clinical experiment. We know what happens to the control group who are cremated and buried. We don’t know what will happen to the experimental group who have opted for cryonics. But we’ve got their DNA, so let’s see what happens.”

As for the sceptics, Kekich has a cunning plan to win them over. “I like to advise proponents of cryonics to stay alive as long as they can,” he says, “so they can make as many converts as possible.”